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| Mirrors > Home > MPE Home > Th. List > sucexeloni | Structured version Visualization version GIF version | ||
| Description: If the successor of an ordinal number exists, it is an ordinal number. This variation of onsuc 7831 does not require ax-un 7755. (Contributed by BTernaryTau, 30-Nov-2024.) (Proof shortened by BJ, 11-Jan-2025.) |
| Ref | Expression |
|---|---|
| sucexeloni | ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eloni 6394 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 2 | ordsuci 7828 | . . 3 ⊢ (Ord 𝐴 → Ord suc 𝐴) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ On → Ord suc 𝐴) |
| 4 | elex 3501 | . 2 ⊢ (suc 𝐴 ∈ 𝑉 → suc 𝐴 ∈ V) | |
| 5 | elong 6392 | . . 3 ⊢ (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴)) | |
| 6 | 5 | biimparc 479 | . 2 ⊢ ((Ord suc 𝐴 ∧ suc 𝐴 ∈ V) → suc 𝐴 ∈ On) |
| 7 | 3, 4, 6 | syl2an 596 | 1 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ 𝑉) → suc 𝐴 ∈ On) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 Ord word 6383 Oncon0 6384 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: onsuc 7831 1on 8518 2on 8520 |
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